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| Main Authors: | , , |
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| Format: | Preprint |
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2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2406.11690 |
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| _version_ | 1866917216140656640 |
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| author | Karabash, Illya M. Lienstromberg, Christina Velázquez, Juan J. L. |
| author_facet | Karabash, Illya M. Lienstromberg, Christina Velázquez, Juan J. L. |
| contents | In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_11690 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Multi-parameter Hopf bifurcations of rimming flows Karabash, Illya M. Lienstromberg, Christina Velázquez, Juan J. L. Analysis of PDEs Dynamical Systems Spectral Theory 35B32 (primary) 76A20, 37L10, 76U05, 35B10, 35K55, 35K25 (secondary) In order to investigate the emergence of periodic oscillations of rimming flows, we study analytically the stability of steady states for the model of (Benilov, Kopteva, O'Brien, 2005), which describes the dynamics of a thin fluid film coating the inner wall of a rotating cylinder and includes effects of surface tension, gravity, and hydrostatic pressure. We apply multi-parameter perturbation methods to eigenvalues of Fréchet derivatives and prove the transition of a pair of conjugate eigenvalues from the stable to the unstable complex half-plane under appropriate variations of parameters. In order to prove rigorously the corresponding branching of periodic solutions from critical equilibria, we extend the multi-parameter Hopf-bifurcation theory to the case of infinite-dimensional dynamical systems. |
| title | Multi-parameter Hopf bifurcations of rimming flows |
| topic | Analysis of PDEs Dynamical Systems Spectral Theory 35B32 (primary) 76A20, 37L10, 76U05, 35B10, 35K55, 35K25 (secondary) |
| url | https://arxiv.org/abs/2406.11690 |